Ratio — Mathslive.ie

Section 1 of 4

Equal Intercepts

When parallel lines cross two transversals, they cut them into pieces in the same ratio. Equal pieces on one line give equal pieces on the other.

The key fact

If $|AB| = |BC|$ on one transversal, then $|DE| = |EF|$ on the other.

Parallel lines cut two transversals. $|AB| = |BC|$ and $|DE| = 12$. Find $|EF|$.

Equal pieces on the left force equal pieces on the right

$|EF| = |DE|$

$|EF| = 12$

On one transversal the pieces are $x+3$ and $12$, and the matching pieces on the other are equal. Find $x$.

Equal pieces: $x + 3 = 12$

$x = 9$

Section 2 of 4

A line parallel to one side of a triangle divides the other two sides in the same ratio.

The ratios

$\dfrac{|AP|}{|PB|} = \dfrac{|AQ|}{|QC|}$ and $\dfrac{|AP|}{|AB|} = \dfrac{|AQ|}{|AC|} = \dfrac{|PQ|}{|BC|}$.

In a triangle a line parallel to the base gives $\dfrac{x}{6} = \dfrac{2}{3}$. Find $x$.

Cross-multiply: $3x = 12$

$x = 4$

Section 3 of 4

You try

In $\triangle ABC$, $PQ \parallel BC$ with $|AP|=5$, $|PB|=12$, $|QC|=15$ and $|PQ|=6$. Find $|AQ|$ and $|BC|$.

Pen and paper out — try it before you reveal.

$\dfrac{|AQ|}{|QC|} = \dfrac{|AP|}{|PB|}$: $\dfrac{x}{15} = \dfrac{5}{12} \Rightarrow 12x = 75 \Rightarrow x = \dfrac{25}{4}$

$\dfrac{|PQ|}{|BC|} = \dfrac{|AP|}{|AB|}$: $\dfrac{6}{y} = \dfrac{5}{17} \Rightarrow 5y = 102$

$|AQ| = \dfrac{25}{4} = 6.25$, $|BC| = \dfrac{102}{5} = 20.4$

Section 4 of 4

You try

A small triangle sits inside a larger one with a line parallel to the base, giving $\dfrac{x}{12} = \dfrac{3}{8}$. Find $x$.

Pen and paper out — try it before you reveal.

Cross-multiply: $8x = 36$

$x = 4.5$

Equal intercepts, a line parallel to one side, and the proportional sides of similar triangles — the whole of Junior Cert ratio in geometry.